Negative refraction and Left-handed behavior in Photonic Crystals:

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Negative refraction and Left-handed behavior in
Photonic Crystals FDTD and Transfer matrix
method studies

Peter Markos, S. Foteinopoulou and C. M. Soukoulis
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Outline of Talk

• What are metamaterials?
• Historical review Left-handed Materials
• Results of the transfer matrix method
• Determination of the effective refractive index
• Negative n and FDTD results in PBGs (ENE SF)
• New left-handed structures
• Experiments on negative refractions (Bilkent)
• Applications/Closing Remarks

E. N. Economou S. Foteinopoulou
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What is an Electromagnetic Metamaterial?

• A composite or structured material that exhibits
  properties not found in naturally occurring
  materials or compounds.
• Left-handed materials have electromagnetic
  properties that are distinct from any known
  material, and hence are examples of metamaterials.

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Electromagnetic Metamaterials
Example Metamaterials based on repeated cells
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Veselago
We are interested in how waves propagate through
various media, so we consider solutions to the
wave equation.
Sov. Phys. Usp. 10, 509 (1968)
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Left-Handed Waves

• If then is a right set of vectors
• If then is a left set of vectors

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Energy flux in plane waves

• Energy flux (Pointing vector)
• Conventional (right-handed) medium
• Left-handed medium

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Frequency dispersion of LH medium

• Energy density in the dispersive medium
• Energy density W must be positive and this
  requires
• LH medium is always dispersive
• According to the Kramers-Kronig relations
• it is always dissipative

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Reversal of Snells Law
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Focusing in a Left-Handed Medium
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• PBGs as Negative Index Materials (NIM)
• Veselago Materials (if any) with e lt 0 and
  mlt 0
• e?m?gt 0 ? Propagation
• k, E, H Left Handed (LHM) ? Sc(E x H)/4p
• opposite to k
•    Snells law with lt 0 (NIM)
•    ?g opposite to k
•    Flat lenses
•    Super lenses

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           Objections to the
left-handed ideas
Parallel momentum is not conserved
Causality is violated
Fermats Principle ?
ndl minimum (?)
Superlensing is not possible
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Reply to the objections

• Photonic crystals have practically zero
  absorption
• Momentum conservation is not violated
• Fermats principle is OK
• Causality is not violated
• Superlensing possible but limited to a cutoff kc
  or 1/L

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Materials with e?lt 0 and m lt0
Photonic Crystals
opposite to
opposite to
opposite to
opposite to
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Super lenses
is imaginary

• Wave components with decay, i.e. are lost , then
  Dmax ? l

If n lt 0, phase changes sign
if
imaginary
thus
ARE NOT LOST !!!
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Metamaterials Extend Properties
J. B. Pendry
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First Left-Handed Test Structure
UCSD, PRL 84, 4184 (2000)
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Transmission Measurements
Transmitted Power (dBm)
6.0
6.5
7.0
5.5
5.0
4.5
Frequency (GHz)
UCSD, PRL 84, 4184 (2000)
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A 2-D Isotropic Structure
UCSD, APL 78, 489 (2001)
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Measurement of Refractive Index
UCSD, Science 292, 77 2001
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Measurement of Refractive Index
UCSD, Science 292, 77 2001
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Measurement of Refractive Index
UCSD, Science 292, 77 2001
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Transfer matrix is able to find

• Transmission (p---gtp, p---gts,) p polarization
• Reflection (p---gtp, p---gts,) s
  polarization
• Both amplitude and phase
• Absorption

Some technical details

• Discretization unit cell Nx x Ny x Nz up to
  24 x 24 x 24
• Length of the sample up to 300 unit cells
• Periodic boundaries in the transverse direction
• Can treat 2d and 3d systems
• Can treat oblique angles
• Weak point Technique requires uniform
  discretization

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Structure of the unit cell
EM wave propagates in the z -direction
Periodic boundary conditions are used in
transverse directions Polarization p wave E
parallel to y s wave E
parallel to x For the p wave, the resonance
frequency interval exists, where with Re meff lt0,
Re eefflt0 and Re np lt0. For the s wave, the
refraction index ns 1.
Typical size of the unit cell 3.3 x 3.67 x 3.67
mm
Typical permittivity of the metallic components
emetal (-35.88 i) x 105
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Structure of the unit cell
SRR
EM waves propagate in the z-direction. Periodic
boundary conditions are used in the xy-plane
LHM
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Left-handed material array of SRRs and wires
Resonance frequency as a function of metallic
permittivity
? complex em
? Real em
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Dependence of LHM peak on metallic permittivity
The length of the system is 10 unit cells
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Dependence of LHM peak on metallic permittivity
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PRB 65, 033401 (2002)
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Example of Utility of Metamaterial
The transmission coefficient is an example of a
quantity that can be determined simply and
analytically, if the bulk material parameters are
known.
UCSD and ISU, PRB, 65, 195103 (2002)
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Effective permittivity e(w) and permeability
m(w) of wires and SRRs
UCSD and ISU, PRB, 65, 195103 (2002)
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Effective permittivity e(w) and permeability
m(w) of LHM
UCSD and ISU, PRB, 65, 195103 (2002)
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Effective refractive index n(w) of LHM
UCSD and ISU, PRB, 65, 195103 (2002)
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Determination of effective parameters from
transmission studies
From transmission and reflection data, the index
of refraction n was calculated. Frequency
interval with Re nlt0 and very small Im n was
found.
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???
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Another 1D left-handed structure
Both SRR and wires are located on the same side
of the dielectric board. Transmission depends on
the orientation of SRR.
Bilkent ISU APL 2002
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0.33 mm
w
tw
t0.5 or 1 mm w0.01 mm
t
0.33 mm
3 mm
l9 cm
0.33 mm
3 mm
ax
Periodicity ax5 or 6.5 mm ay3.63 mm az5
mm Number of SRR Nx20 Ny25 Nz25
Polarization TM
y
E
x
y
x
z
B
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New designs for left-handed materials
eb4.4
Bilkent and ISU, APL 81, 120 (2002)
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ax6.5 mm t 0.5 mm
Bilkent ISU APL 2002
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ax6.5 mm t 1 mm
Bilkent ISU APL 2002
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Cut wires Positive and negative n
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Phase and group refractive index

• In both the LHM and PC literature there is still
  a lot of confusion regarding the phase refractive
  index np and the group refractive index ng. How
  these properties relate to negative refraction
  and LH behavior has not yet been fully examined.
• There is controversy over the negative
  refraction phenomenon. There has been debate
  over the allowed signs ( /-) for np and ng in
  the LH system.

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DEFINING phase and group refractive index np and
ng

• In any general case
• The equifrequency surfaces (EFS) (i.e. contours
  of constant frequency in 2D k-space) in air and
  in the PC are needed to find the refracted
  wavevector kf (see figure).
• vphasec/np and vgroup
  c/ng
• Where c is the velocity of light
• So from k// momentum conservation npc kf
  (?) /?.

Remarks

• In the PC system vgroupvenergy so nggt1.
  Indeed this holds !
• np lt1 in many cases, i.e. the phase velocity is
  larger than c in many cases.
• np can be used in Snells formula to determine
  the angle of the propagating wavevector. In
  general this angle is not the propagation angle
  of the signal. This angle is the propagation
  angle of the signal only when dispersion is
  linear (normal), i.e. the EFS in the PC is
  circular (i.e. kf independent of theta).
• ng can never be used in a Snell-like formula to
  determine the signals propagation angle.

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Index of refraction of photonic crystals

• The wavelength is comparable with the period of
  the photonic crystal
• An effective medium approximation is not valid

Equifrequency surfaces
Effective index
Refraction angle
kx
ky
w
Incident angle
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Photonic Crystals with negative refraction.
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Photonic Crystals with negative refraction.
S. Foteinopoulou, E. N. Economou and C. M.
Soukoulis
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Schematics for Refraction at the PC interface
EFS plot of frequency a/l 0.58
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Schematics for Refraction at the PC interface
EFS plot of frequency a/l 0.535
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Negative refraction and left-handed behavior for
a/l 0.58
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Negative refraction but NO left-handed behavior
for a/l 0.535
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Superlensing in 2D Photonic Crystals
Lattice constant4.794 mm Dielectric
constant9.73 r/a0.34, square lattice
Experiment by Ozbays group
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Negative Refraction in a 2d Photonic Crystal
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Band structure, negative refraction and
experimental set up
Frequency13.7 GHz
Negative refraction is achievable in this
frequency range for certain angles of incidence.
Bilkent ISU
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Superlensing in photonic crystals
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Subwavelength Resolution in PC based Superlens
The separation between the two point sources is
l/3
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Photonic Crystals with negative refraction.
Photonic Crystal
vacuum
FDTD simulations were used to study the time
evolution of an EM wave as it hits the interface
vacuum/photonic crystal. Photonic crystal
consists of an hexagonal lattice of dielectric
rods with e12.96. The radius of rods is
r0.35a. a is the lattice constant.
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Photonic Crystals with negative refraction.
t01.5T Tl/c
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Photonic Crystals with negative refraction.
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Photonic Crystals with negative refraction.
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Photonic Crystals with negative refraction.
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Photonic Crystals negative refraction
The EM wave is trapped temporarily at the
interface and after a long time, the wave front
moves eventually in the negative direction.
Negative refraction was observed for wavelength
of the EM wave l 1.64 1.75 a (a is the
lattice constant of PC)
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Conclusions

• Simulated various structures of SRRs LHMs
• Calculated transmission, reflection and
  absorption
• Calculated meff and eeff and refraction index
  (with UCSD)
• Suggested new designs for left-handed materials
• Found negative refraction in photonic crystals
  
• A transient time is needed for the wave to move
  along the - direction
• Causality and speed of light is not violated.
• Existence of negative refraction does not
  guarantee the existence of
• negative n and so LH behavior
• Experimental demonstration of negative
  refraction and superlensing
• Image of two points sources can be resolved by a
  distance of l/3!!!

DOE, DARPA, NSF, NATO, EU
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• Publications
• P. Markos and C. M. Soukoulis, Phys. Rev. B 65,
  033401 (2002)
• P. Markos and C. M. Soukoulis, Phys. Rev. E 65,
  036622 (2002)
• D. R. Smith, S. Schultz, P. Markos and C. M.
  Soukoulis, Phys. Rev. B 65, 195104 (2002)
• M. Bayindir, K. Aydin, E. Ozbay, P. Markos and
  C. M. Soukoulis, Appl. Phys. Lett. (2002)
• P. Markos, I. Rousochatzakis and C. M.
  Soukoulis, Phys. Rev. E 66, 045601 (R) (2002)
• S. Foteinopoulou, E. N. Economou and C. M.
  Soukoulis, PRL, accepted (2003)
• S. Foteinopoulou and C. M. Soukoulis, submitted
  Phys. Rev. B (2002)
• P. Markos and C. M. Soukoulis, submitted to Opt.
  Lett.
• E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou
  and C. M. Soukoulis, submitted to Nature
• P. Markos and C. M. Soukoulis, submitted to
  Optics Express

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The keen interest to the topic

• Terminology

• Left-Handed Medium (LH)
• Metamaterial
• Backward Medium (BW)
• Double Negative Medium (DNG)
• Negative Phase Velocity (NPV)
• Materials with Negative Refraction (MNR)

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Collaboration between Crete, Greece and Bilkent
University, Turkey
Crete
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GM
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Image Plane
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Experimental Setup
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Scanned Power Distribution at the Image Plane
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Dependence of LHM peak on L and Im em
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Dependence on the incident angle
Transmission peak does not depend on the angle
of incidence !
Transition peak strongly depends on the angle of
incidence.
This structure has an additional xz - plane of
symmetry
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Transmission depends on the orientation of SRR
Transmission properties depend on the orientation
of the SRR

• Lower transmission
• Narrower resonance interval
• Lower resonance frequency

• Higher transmission
• Broader resonance interval
• Higher resonance frequency

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Dependence of the LHM T peak on the Im eBoard
In our simulations, we have Periodic boundary
condition, therefore no losses due to scattering
into another direction. Very high Im emetal
therefore very small losses in the
metallic components.
Losses in the dielectric board are crucial for
the transmission properties of the LH structures.
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New / Alternate Designs
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Superprism Phenomena in Photonic Crystals
Experiment

• H.Kosaka, T.Kawashima et. al. Superprism
  phenomena in photonic crystals, Phys. Rev. B 58,
  10096 (1998)

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Scattering of the photonic crystalHexagonal 2D
photonic crystal

• M.Natomi, Phys. Rev. B 62, 10696 (2000)
• Using an equifrequency surface (EFS) plots

Vanishingly small index modulation
Small index modulation
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Photonic crystal as a perfect lens
C. Luo, S. G. Johnson, J. D. Joannopoulos, and J.
B. Pendry Phys. Rev. B, 65, 201104 (2002)
Resolution limit 0.67 l
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